An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold
(
Z
)
{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
Hi,
I am looking for an explanation, if any, on why every integer square leaves remainder 0 or 1 on division by 4.
Appreciate your time and help
bluemoon2188
Homework Statement
Find the numbers, if any, where the function is discontinuous.
f(x) = [[x - 2]]
The attempt at a solution
function is discontinuous for all integer values of x.
I know that this is the obvious answer, however I am required to explain this in clear mathematical...
Homework Statement
Consider the following puzzle. You are to choose 4 three-letter "words" from the following list:
ECB EFH BEJ GGE HIJ CDE GEG CBJ
For each word, you earn a score equal to the position that the word's third letter appears in the alphabet. For example, ECB earns a score...
I realize that this might seems to be a strange question, but after doing some coding i realized the following.
to brute force the factorization of all numbers less than one million takes around 665 million tests (i.e. does this number divide the original).
to do it "smarter" (least i...
I'm not sure at all how best to approach this problem. Basically, the application is for practicing mental arithmetic. Most of the features are pretty easy, with the exception of division, which is the topic here.
I accept a minimum and maximum for the first and last numbers in the division...
"assignment makes pointer from integer without cast"
elephant*
get_elephants()
{
elephant *current, *first;
int response;
/* create first node */
first = (elephant*)calloc(1,sizeof(elephant)); /* THIS LINE */
current = first...
I am trying to understand how I can find the square root of a large prime number in the form of an integer value, the portion after the decimal is irelevant.
The numbers I wish to compute range around 188 multiplied by 3 to the power of 6548 plus 1, as an example. so let's say in excess of...
Find the form of all n in N satifying tow(n)=6. (sorry don't know how to write this Tex). What is the smallest positive integer n for which this is true?
tow(n)=6 so 6=(1+a1)(1+a2)----(1+ak) where n=p1^a1p2^a2---pk^ak
Show that if n is a positive integer, then C(n,0) < C(n,1) <...<C(n, floor(n/2)) = C
Homework Statement
Show that if n is a positive integer, then C(n,0) < C(n,1) <...<C(n, floor(n/2)) = C(n, ceiling(n/2)) > ... > C(n, n)
Homework Equations
none
The Attempt at a Solution
Seems pretty...
Homework Statement
If a \in Z, then a^3 \equiv a (mod 3) .
2. The attempt at a solution
Proof: Suppose a \in Z. Thus a is either odd or even.
Case 1: Let a be even. Thus a = 2k, for some k \in Z. So a^3 - a = 8k^3 -2k = 2(4k^3 - k) = 2(k)(2k - 1)(2k + 1). Notice that, for all k...
I know that the fundamental theorem of arithmetic states that any integer greater than 1 can be written as an unique prime factorization.
I was wondering if there is any concept of negative prime numbers, because any integer greater than 1 or less than -1 should be able to be written as n = p1...
Hi guys, i am using the C language and have created the following function:
void counter (double c) {
FILE *fp;
char output[]="output.xls";
int n,p=0;
int r=0;
int i,j;
float x=0;
fp=fopen(output,"w");
fprintf(fp,"rmax\tnumber of particles within rmax\r\r\n");
for(n=1;n<=10;n++)...
Number theory: confused about the phrase "an integer of the form"
Homework Statement
Prove that any prime of the form 3k+1 is of the form 6k+1
Homework Equations
The Attempt at a Solution
I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But...
Hello everyone!
I am trying to construct an algorithm for the following problem and was wondering if there is any existing body of knowledge on this. Please forgive me if this is inappropriate (or ridiculous) but I am totally foreign to number theory.
It goes like this:
You are given n...
Beaconaut APICalc 2 just released on Jan.18, 2011, which is an arbitrary-precision integer calculator for bignum arithmetic, cryptography analysis and number theory research...
http://www.beaconaut.com/forums/default.aspx?g=posts&t=13"
Hello everyone
Thanks for viewing this topic for me... my question is when I tried to check user inputs a valid age it works only if a number is entered. If by mistake user inputs a character value (e.g. any character a to z or any symbol) this loop runs infinite
How can I check if a...
Homework Statement
I can't find a step by step explanation for solving these types of equations
eg.
99 = [2x+1]/3
Homework Equations
eg.
99 = [2x+1]/3
or
48 = 4[2x/3]
How do you handle the multipliers iand constants inside the brackets?
thx
[b]3. The Attempt at a...
Homework Statement
This is just a problem i came up with while doing in equations, and recently saw in a book too.
Find the smallest integer n, for which
n! > 10n
The Attempt at a Solution
The trial method (with help from the computer) yields 25 to be the answer.
After a...
Hi everybody!
I'm studying the IQHE and I want to understand the rise of the diophantine equation. I read the thouless article but it was no so conclusive.
I've also read the kohmoto article (phys rev B 1989) and he says that that property comes from the darboux theorem but i don't...
Homework Statement
This is a proof problem in the mathematical induction section of my textbook. I am having trouble with question (c).
Result: 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6 for every positive integer n.
(a) Use the result to determine the formula for 22 + 42 + 62 + ... + (2n)2 for...
Homework Statement
Suppose that n is an odd integer. Prove that n is either one greater than a multiple of 4 or one less than a multiple of 4.
Homework Equations
N/A
The Attempt at a Solution
I realize that this is going to be a direct proof. However, I am stumped on where to...
Homework Statement
For some integer n, a|(4n+3) and a|(2n+1). Therefore, 4n+3 is an integer multiple of a, as well as (2n+1). Prove or disprove that a=+/-1.
Homework Equations
N/A
The Attempt at a Solution
I have been working on this one for quite some time now, but I cannot...
Homework Statement let a,b be positive integer , c is real number, and -a<c<b
i want to show there exist integer m, -a \leq m \leq b such that m-1 \leq c<m
i don't know any easy method, but this is where i got now,
Let set S=[m|-a \leq m \leq b]
So by contradiction,
suppose that for all m...
When looking for potential factors of an integer, I noticed that I could predict the frequency of the greatest integer function quotient and use this prediction to jump to the next potential factor. See the attachment for an example. I don't know if someone else had already discovered this...
Homework Statement
Let k be any positive integer. Prove that there exists a positive
integer multiple n of k such that the only digits in n are 0s and
1s. (Use the pigeonhole principle.)
Homework Equations
The General Pigeonhole Principle
If more than mk things are distributed into k...
Hey there, physics forums!
A question occurred to me the other day: Is it true that if f \in \mathbb{Z}[x] is monic and irreducible over \mathbb{Q} , then for at least one a \in \mathbb{Z} , f(a) is prime? I can't prove it, but I suspect it's true. Does anyone know if this problem...
Homework Statement
√(3+2√(2)) can be simplified into a fairly simple sum of two number: one is an integer and one is an irrational number . Find them and show you are correct by squaring both sides of the "equation"
Homework Equations
The Attempt at a Solution
Hello PhysicsForums!
I was reading up on UFD's and I came up with a few quick questions.
1. Why don't the integers of Q[\sqrt{-5}] form a UFD? I was trying to tie in the quadratic integers that divide 6 to help me understand this, but I am stuck.
2. Why is Z a UFD?
3. Assuming Q[\sqrt{d}] is...
Homework Statement
I want to prove that a polynomial f(x) and a polynomial g(x) with degrees of k,n where k,n are positive even integer, n>k
that limit x-> - infinity of f(x)-g(x)=-infinity
Homework Equations
a polynomial can be written as
a1x^n+a2x^(n-1)...+a(n-1)x+an
The...
Homework Statement
Prove that either [a]_{n}\cap[b]_{n}=empty set or [a]_{n}=[b]_{n}.Homework Equations
The Attempt at a Solution
I want to assume there is an element x in [a]_{n}\cap[b]_{n} and show this implies [a]_{n}=[b]_{n}.
This tells me x is in [a]_{n} and [b]_{n}.
That's where I get stuck.
Homework Statement
For what values of the constant K does the differential equation
y"-(1/4 +K/x)y=0 (0<X< infinity)
have a nontrival solution vanishing at x=0 and x= infinity ?
Homework Equations
Hints that were suggested for my prof were:
For large x...
I have 2 questions here. I know there is a lot of text, but don;t be scared or deterred from helping please, it is simple to understand and mostly just text/explanation. The Second one is very long, the first not so much.
1)Problem: Complete the proof of Observation 4.2 by showing that every...
Homework Statement
find p,q,r,s integer solution
\frac{1}{p^2}+\frac{1}{q^2}+\frac{1}{r^2}+\frac{1}{s^2}=1
Homework Equations
here some alternate form you can see
http://www.wolframalpha.com/input/?i=1/p^2%2B1/q^2%2B1/r^2%2B1/s^2%3D1
The Attempt at a Solution
i don't know...
Homework Statement
let x,y \in N. Solve the equation \frac{1}{x}+\frac{1}{y}=1
Homework Equations
n/a
The Attempt at a Solution
so i can see x=y=2 is the solution, hmm, isn't there any other solution.
and, this is one of three of my assignment question for 15% continuous...
Homework Statement
prove that if a and b are both odd integer, then 16|(a^2+b^2-2)
Homework Equations
n/a
The Attempt at a Solution
let a=2m+1 \ and \ b=2n+1, then a^2+b^2-2=4(m(m+1)+n(n+1)) so its divisible by 4, and also divisible 8 since m(m+1) \ and \ n(n+1) are even.
so i only prove...
In the situation where differences between consecutive squares, (or consecutive cubes, consecutive x^4, etc.) are calculated,
then the differences between those differences are calculated, and then the differences of those differences, and so on until you reach a constant number at a deep...
Hi all,
I am new to topology & geometry. I skimmed through the subject in various books so I might be asking simple questions.
My question is how can we say Chern's class belong to Integer rather than Real cohomology class? I assume one defines it through the characteristic polynomial of...
Homework Statement
n2 has the form 3k or 3k+1 for some integer k
Homework Equations
n/a
The Attempt at a Solution
i've tried to table it for me to see, but still i don't have the general idea, should i show n2 is either divisable by 3k or 3k+1? if yes i still don't know owho
Please help - I am helping a student who is trying to translate Ong-Schnorr to C++ or VB.net language. My major was electronics so I am not sure how to interpret inverse random integer k. Any one ? or suggest me a reference to read on - much thanks, newbie
Ong-Schnorr-Shamir (from Briuce's...
A problem asks to find all possible pairs (x, y) of positive integers that satisfy the equation:
x3 = y2 – 15
There are 2 pairs (so far) that satisfy the equation:
x = 1, y = 4
x = 109, y = 1138
It's possible that these 2 points are the only two positive integer solutions...
I am struggling to understand the proof for integer parts of real numbers. I have used to mean less than or equal to because I could not work out how to type it in. I need to show that:
∃ unique n ∈ Z s.t. nx<n+1
The proof given is the following:
Let
A={k∈Z : kx}
This is a...
I am attempting to program Mathematica to multiply the square free terms of an integer. Basically say we are looking at 252, its prime factors are 2^2*3^2*7. So what I want to do is have Mathematica return to me just 2*3*7 when I enter 252.
So I have this
S := FactorInteger[252]...